Seminar Analysis: Xavier Ros-Oton (Polytechnic University of Catalonia)

Boundary regularity for elliptic integro-differential equations

We study the boundary regularity of solutions to elliptic integro-differential equations. First we prove that, for the fractional Laplacian (-Δ)^s with s∈(0,1), solutions u satisfy that u/d^s is Hölder continuous up to the boundary, where d(x) is the distance to the boundary of the domain Ω. We will show that, in this fractional context, the quantity u/d^s|_∂Ω plays the role that the normal derivative plays in second order equations. Finally, we also present new boundary regularity results for fully nonlinear integro-differential equations.

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